SOLUTION: Water is poured into a conical paper cup at the rate of 3/2 in^3/sec . If the cup is 6 inches tall and the top has a radius of 3 inches, how fast is the water level rising when th

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Question 994582: Water is poured into a conical paper cup at the rate of 3/2 in^3/sec . If the cup is 6 inches tall and the top
has a radius of 3 inches, how fast is the water level rising when the
water is 3 inches deep?
The water level is rising at a rate of____
Thank you
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The volume of a cone of height h and radius R is given by
V=%281%2F3%29%2Api%2AR%5E2%2Ah .
I am not going to keep mentioning units,
but time, t is understood to be measured in seconds;
h and R are understood to be measured in inches,
and V is understood to be in cubic inches.
The cup is filling at a rate of dV%2Fdt=3%2F2 .
As the water is filling the conical paper cup,
the water is in the shape of a similar cone,
with a radius to height ratio of 3%2F6=1%2F2 ,
so R=h%2F2 .
The volume of water in the cup as a function of the water-cone height is
.
The inverse function is


When h=3 , V=%28pi%2F12%29%2A3%5E3=%28pi%2F12%29%2A9=9pi%2F4 ;
, and
dh%2Fdt=%28dh%2FdV%29%2A%28dV%2Fdt%29=%284%2F9pi%29%2A%283%2F2%29=highlight%282%2F3pi%29